In this thesis we are interested in (unknown) functions which appear in statistical models, and testing procedures concerning these unknown functions. These functions are estimated flexibly (nonparametric) and not according to a prespecified (parametric) form. The nonparametric technique we consider is by spline approximations. Splines are used to estimate univariate as well as multivariate functions. Then, hypothesis testing about those unknown functions is translated to a testing procedure based on the spline estimation.

The first statistical model we consider is a varying coefficient model (VCM), which is an extension of the classical linear regression model in the sense that the regression coefficients are allowed to be functions, for example of time. Varying coefficient models (VCMs) are since many years popular in longitudinal data and panel data studies, and have been applied in fields such as finance, economics, ecology, epidemiology, health sciences, and so on. We estimate the coefficient by B-splines. An important question in a VCM is whether the coefficient has a particular parametric form, such as being constant or linear. This allows, on the one hand to draw conclusions on the effect of certain variables on the response variable. On the other hand, this could allow to propose a simpler model and strongly reduce the number of parameters in the model. We construct testing procedures to answer the former hypothesis, and give the supporting theoretical results for longitudinal data with correlated errors. Testing of such hypotheses in VCMs is studied in Chapter 2 with illustrations of the power through simulations and a data application.

In Chapter 3 we address our second hypothesis of VCMs. There, we are interested in whether a coefficient function is monotonic or convex, i.e. the shape. We develop testing procedures for monotonicity and convexity, with the necessary theoretical results. Moreover, we give procedures to test simultaneously the shapes of certain coefficient functions. The tests use constrained and unconstrained regression splines. Application of our testing procedures on simulations reveal the effectiveness of our approach. Data applications are also given.

Chapter 4 studies parameters of partial differential equations (PDEs). Many complex dynamic systems are governed by PDEs, they appear in a vast number of scientific fields such as biology, physics and finance. PDEs are determined by their parameters. Often scientists face the challenge to determine unknown parameters of a PDE, and the need to estimate them from error prone measurements. In the statistical literature it is very often assumed that the parameters are constant, which restricts the application possibilities because in reality this assumption can be crude. In Chapter 4 we extend the parametric cascading method- which was shown to be effective for PDE models with constant parameters- to the PDE setting where the coefficients vary with multiple variables. In the case of a linear PDE model, we show that our proposed estimator of the parameters is uniformly consistent.

In Chapter 1 we introduce further the concepts of this thesis with an overview of the relevant statistical literature. Finally, in Chapter 5 we conclude this thesis with a summary of the results and discuss future research perspectives.